Equiconsistencies at subcompact cardinals (English)

In: Archive for mathematical logic   ;  55 ,  1  ;  2016
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We present equiconsistency results at the level of subcompact cardinals. Assuming SBH δ , a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □(δ) and □ δ fail, then δ is subcompact in a class inner model. If in addition □(δ +) fails, we prove that δ is $${\Pi_1^2}$$ Π 1 2 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBH δ holds, the Proper Forcing Axiom implies the existence of a class inner model with a $${\Pi_1^2}$$ Π 1 2 subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ +(n) supercompact for all n <� ω. We state some results at this level, and indicate how they are proved.

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Dedicated to Richard Laver
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